Pattern avoiding alternating involutions

TL;DR

This paper studies the enumeration and characterization of alternating involutions avoiding specific patterns of length three or four, revealing connections to well-known sequences like Motzkin and Fibonacci numbers.

Contribution

It provides new insights into pattern avoidance in alternating involutions, especially for length four patterns, linking to classical combinatorial sequences.

Findings

Length three pattern avoidance is trivial.

Length four pattern avoidance involves Motzkin and Fibonacci numbers.

New characterizations of certain involution classes.

Abstract

We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length four case is more challenging and involves sequences of combinatorial interest, such as Motzkin and Fibonacci numbers.